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We consider the contact between an elastic body and a deformable foundation. Firstly, we introduce a mathematical model for this phenomenon by means of a normal compliance contact condition associated with a friction law. Then, we propose a variational formulation of the model in a form of a quasi-variational inequality governed by a non-differentiable functional and we briefly discuss its well-possedness. Nextly, we address an optimal control problem related to this model in order to led the displacement field as close as possible to a given target by acting with a localized boundary control. By using some mollifiers of the normal compliance functions, we introduce a regularized model which allows us to establish an optimality condition. Finally, by means of asymptotic analysis tools, we show that the solutions of the regularized optimal control problems converge to a solution of the initial optimal control problem.

NOABSTRACTThe purpose of this study is to present the first Romanian case-series of patients with heart failure with reduced ejection fraction (HFrEF), supported with the newest generation of cardiac contractility modulation (CCM) device.16 patients (15 men), aged 66.6±7.49 years, were supported with OPTIMIZER® smart IPG CCMX10 device and followed-up for an average duration of 385.75±326.32 days. The etiology of HF was ischemic in 13 patients (81%), 8 patients (50%) had atrial fibrillation, mean creatinine clearance value was 55.8±13.87 ml/min, and 5 patients (31,2%) had diabetes mellitus. All patients were supported with an implanted cardio verter-defibrillator (ICD), while 5 patients (31.2%) had cardiac resynchronization therapy (CRT) on top. The pharma cological treatment has been optimized in all patients. Six months after implantation, the LVEF has increased from 25.93%±6.21 to 35.5%±4.31 (p=0.00002), NYHA class improved from 3.18±0.4 to 1.83±0.38 (p<0.0001), and exercise tolerance evaluated with 6-minute walking test (6MWT) increased (from 321.87±70.63m to 521.41±86.43m; p<0,00001). Three patients (18,7%) died during the follow-up period after 48, 108 and 545 days (one non-cardiac death).Cardiac contractile therapy is a feasible, safe, and useful therapy for patients with HFrEF whose symptomatology is not improved with optimal standard therapy.

This article deals with the approximation of the boundary control of the linear one-dimensional wave equation. It is known that the high frequency spurious oscillations that the classical methods of finite difference and finite element introduce lead to nonuniform controllability properties (see [J. A. Infante and E. Zuazua, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 407-438]. A space-discrete scheme with an added numerical vanishing viscous term is introduced and analyzed. The extra numerical damping filters out the high numerical frequencies and ensures the convergence of the sequence of discrete controls to a control of the continuous conservative wave equation when the mesh size tends to zero.

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate boundary controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.

(1) Background: Heart failure (HF) is a major cause of morbidity and mortality throughout the world. Despite substantial progress in its prevention and treatment, mortality rates remain high. Device therapy for HF mainly includes cardiac resynchronization therapy (CRT) and the use of an implantable cardioverter-defibrillator (ICD). Recently, however, a new device therapy—cardiac contractility modulation (CCM)—became available. (2) Aim: The purpose of this study is to present a first case-series of patients with different clinical patterns of HF with a reduced ejection fraction (HFrEF), supported with the newest generation of CCM devices. (3) Methods and results: Five patients with a left ventricular ejection fraction (LVEF) ≤ 35% and a New York Heart Association (NYHA) class ≥ III were supported with CCM OPTIMIZER® SMART IPGCCMX10 at our clinic. The patients had a median age of 67 ± 8.03 years (47–80) and were all males—four with ischemic etiology dilated cardiomyopathy. In two cases, CCM was added on top of CRT (non-responders), and, in one patient, CCM was delivered during persistent atrial fibrillation (AF). After 6 months of follow-up, the LVEF increased from 25.4 ± 6.8% to 27 ± 9%, and the six-minute walk distance increased from 310 ± 65.1 m to 466 ± 23.6 m. One patient died 47 days after device implantation. (4) Conclusion: CCM therapy provided with the new model OPTIMIZER® SMART IPG CCMX10 is safe, feasible, and applicable to a wide range of patients with HF.

We propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's \"stabilizability implies controllability\" principle with the Galerkin method. The main new feature of this work consists of giving precise error estimates. In order to test the efficiency of the method, we consider two illustrative examples (with the finite element approximations of the wave and the beam equations) and describe the corresponding simulations. [PUBLICATION ABSTRACT]

The null-controllability property of a $1-d$ parabolic equation involving a fractional power of the Laplace operator, $(-\\Delta)^\\alpha$, is studied. The control is a scalar time-dependent function $g=g(t)$ acting on the system through a given space-profile $f=f(x)$ on the interior of the domain. Thus, the control $g$ determines the intensity of the space control $f$ applied to the system, the latter being given a priori. We show that, if $\\alpha\\leq 1/2$ and the shape function $f$ is, say, in $L^2$, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case $\\alpha >1/2$ and, in particular, for the heat equation that corresponds to $\\alpha=1$. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation. On the contrary, if more regularity of the shape function $f$ is assumed, then we show that there are initial data in any Sobolev space $H^m$ that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases. These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S. Micu and E. Zuazua, Trans. Amer. Math. Soc., 353 (2000), pp. 1635-1659] and [S. Micu and E. Zuazua, Portugal. Math., 58 (2001), pp. 1-24]. We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.

We study the boundary controllability properties of the linearized Benjamin--Bona--Mahony equation $$\\left\\{ \\begin{array}{ll} u_t-u_{xxt}+u_x=0,& x\\in(0,1),\\,t > 0,\\\ u(t,0)=0,\\,\\, u(1,t)=f(t),&t>0. \\end{array} \\right. We show that the equation is approximately controllable but not spectrally controllable (no finite linear combination of eigenfunctions, other than zero, is controllable). Next, we prove a finite controllability result and we estimate the norms of the controls needed in this case.

The aim of this paper is to study a boundary time-optimal control problem for the heat equation in a two-dimensional ball. The main ingredient is the extension of a result concerning Müntz polynomials due to Borwein and Erdélyi that allows us to prove an observability inequality for the dynamical system's truncation to a finite number of modes. This result, combined with a well-known Lebeau–Robbiano argument used to show the null-controllability of parabolic type equations, enables us to deduce the existence, uniqueness and bang-bang properties for the boundary time-optimal control.

This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted L 2 -norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.